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arXiv:2605.23228

Exact solution of generalized gauge-invariant Ising chains with multi-spin interactions

Pavel Khrapov, Stepan Shchurenkov

Published May 22, 2026• 0 citations
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Abstract

In this work, exact solutions are obtained for a class of generalized gauge-invariant $n$-chain Ising models ($n=1,2,3,4$) with arbitrary multi-spin interactions that are invariant under the local $\mathbb{Z}_2$ gauge group. On a strip lattice of finite length $L$ and width $n$ with periodic or free boundary conditions, an explicit expression for the partition function is derived using the transfer-matrix method. Two successive transformations are developed: elimination of gauge redundancy and reduction of the original model to an effective $n$-chain Ising model with all possible interactions between neighboring vertical layers. On the basis of the spectral decomposition of the $2^n\times 2^n$ transfer matrix, general formulas are obtained for gauge-invariant correlation functions and Wilson loops of arbitrary width. For $n \le 3$, explicit expressions are derived in terms of eigenvalues and eigenvectors. A detailed analysis of the behavior of the Wilson loop is performed, which allows us to identify regimes exhibiting area-law (confinement-like) and perimeter-law (deconfinement-like) dependence. For specific Hamiltonians, the string tension is computed and the corresponding phase diagrams are constructed. The results generalize and substantially extend the classical works on the gauge-invariant Ising model.

DOI
10.48550/arxiv.2605.23228

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In this work, exact solutions are obtained for a class of generalized gauge-invariant nnn-chain Ising models (n=1,2,3,4n=1,2,3,4n=1,2,3,4) with arbitrary multi-spin interactions that are invariant under the local Z2\mathbb{Z}_2Z2​ gauge group. On a strip lattice of finite length LLL and width nnn with periodic or free boundary conditions, an explicit expression for the partition function is derived using the transfer-matrix method. Two successive transformations are developed: elimination of gauge redundancy and reduction of the original model to an effective nnn-chain Ising model with all possible interactions between neighboring vertical layers. On the basis of the spectral decomposition of the 2n×2n2^n\times 2^n2n×2n transfer matrix, general formulas are obtained for gauge-invariant correlation functions and Wilson loops of arbitrary width. For n≤3n \le 3n≤3, explicit expressions are derived in terms of eigenvalues and eigenvectors. A detailed analysis of the behavior of the Wilson loop is performed, which allows us to identify regimes exhibiting area-law (confinement-like) and perimeter-law (deconfinement-like) dependence. For specific Hamiltonians, the string tension is computed and the corresponding phase diagrams are constructed. The results generalize and substantially extend the classical works on the gauge-invariant Ising model.
In this paper the exact solution and correlation functions for a double-chain Ising model with multi-spin interactions and symmetric Hamiltonian density are obtained. The study employs the transfer matrix method to derive fundamental thermodynamic characteristics of the system. The main results include exact expressions for the partition function, free energy, internal energy, specific heat capacity, magnetization, susceptibility, and entropy in a strip of finite length and in the thermodynamic limit. The work provides explicit formulas for the eigenvalues and shows structure of eigenvectors of the transfer matrix. The expression for magnetization in the thermodynamic limit using components of normalized eigenvector corresponding to the maximum eigenvalue is obtained. A detailed analysis is conducted for a special case of interactions involving all kinds of two- and four-spin interactions. This gives the simplified formula for free energy, it is calculated using the root of quadratic equation. Using special relation between parameters the solution for gonihedric model on the plane is obtained. The research reveals properties of the system, including specific features of ground states and phase diagram characteristics. Particular attention is given to the behavior of physical quantities near frustration points and the investigation of spin correlation functions. Plots of physical characteristics, including inverse correlation length, illustrating the obtained results are constructed.
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We obtain an exact solution for a generalized three-chain Ising tube (TCGIT) of length LLL with toroidal boundary conditions and the most general C3C_3C3​-invariant Hamiltonian on an elementary prism, containing 20 independent coupling constants, including an external magnetic field. Using an 8×88\times 88×8 transfer matrix, we derive the exact partition function of the finite system and obtain the free energy, internal energy, specific heat, magnetization, magnetic susceptibility, and entropy in the thermodynamic limit L→∞L\to\inftyL→∞. In the general case, λmax⁡λ_{\max}λmax​ is determined by a quartic equation, whereas in the principal special case with even-spin interactions (PSC) the spectrum simplifies substantially: the characteristic polynomial factorizes, and λmax⁡λ_{\max}λmax​ is given by the root of a quadratic equation. For mirror-symmetric subfamilies, we derive explicit formulas for the pair correlation functions and express the magnetization in terms of the components of the eigenvector associated with λmax⁡λ_{\max}λmax​; in the even-spin case with h=0h=0h=0, the magnetization vanishes. Important special cases include the width-three planar model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions, including the entropy limit S(T→0+)=(ln⁡2)/3S(T\to0^+)=(\ln 2)/3S(T→0+)=(ln2)/3 for k≥0k\ge 0k≥0 and S(T→0+)=0S(T\to0^+)=0S(T→0+)=0 for k<0k<0k<0, as well as the width-three planar triangular model with distinct nearest-neighbor couplings, three-spin interactions involving neighboring triangles, and an external field.
Abstract These lecture notes deal with Ising models with interactions containing products of more than two spins. After a number of examples have been given, some general statements are presented. Of particular interest is a gauge-invariant Ising model in four dimensions. This has important properties in common with models for quantum chromodynamics as developed by Ken Wilson. One phase yields an area law for the Wilson loop, yielding an interaction that increases proportionally to the distance and thus corresponds to quark confinement. The other phase yields a perimeter law allowing for a quark–gluon plasma.
For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.
The susceptibility in arbitrary directions, the specific heat, the energy and the magnetization of general Ising-type chains are exactly expressed in terms of the eigenvectors of the transfer matrix corresponding to the Ising-type interaction. These quantities, as a result, can be explicitly obtained for small spin values and generally obtained from the solution of corresponding eigenvalue problem of finite degree. Numerical estimations are easy for arbitrary spin values. This formula includes the spin- S transverse Ising model with vanishing or non-vanishing parallel external field, the Blume-Emery-Griffiths model and the Blume-Capel model, mixed spin and mixed bond periodic systems such as alternating Ising chains or Ising ferrimagnets and also includes Ising models with random structures. Low temperature behaviors of these systems and the crossover to infinite spin systems are also investigated.